Finite Group Databasebeta

← #25102240

Subgroups — #25102240

Order 11008.

1952 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583033C211yesmaximal, characteristic
2C2×#14583009C212yesmaximal, complemented
3C4×#13063299C211yesmaximal, characteristic, complemented
4#19134943C212yesmaximal, complemented
5#19134912C211yesmaximal, characteristic, complemented
6#19134856C212yesmaximal, complemented
7#19134788C211yesmaximal, characteristic, complemented
8#19134526C211yesmaximal, characteristic, complemented
9#19134420C212yesmaximal, complemented
10#19134357C211yesmaximal, characteristic, complemented
11C22×C4×D4×C43C211yesmaximal, characteristic, complemented
12C22×#10621882C2212yes—
13C2×#13063299C2211yescharacteristic
14C2×C4×D172C2211yescharacteristic, complemented
15C2×C4×C22⋊D43—21no—
16#14583033C418yescomplemented
17#14583009—28no—
18C2×#13063256C2212yescomplemented
19C2×#13063246C2212yes—
20C2×#13063246C2212yes—
21C2×#13063246C2212yes—
22C2×#13063245C2212yescomplemented
23C2×C22⋊(C43⋊C8)C2211yescharacteristic
24C2×C22⋊(C43⋊C8)C2211yescharacteristic
25C2×#13063236C2212yescomplemented
26#14582801C2211yescharacteristic, complemented
27#14582797C2211yescharacteristic, complemented
28#14582790—21no—
29#14582778—21no—
30#14582759C2211yescharacteristic
31C4×C4⋊Dic43C2211yescharacteristic, complemented
32C4×(C2×D43)⋊C4—21no—
33#14582661C2212yescomplemented
34#14582661C2212yescomplemented
35#14582654C2212yescomplemented
36C23×D4×C43C2211yescharacteristic
37C24×C4×C43—21no—
38C2×C4×D4×C43—28no—
39C2×C4×D4×C43—21no—
40C2×C4×D4×C43—21no—
41C2×C4×D4×C43C2212yes—
42C2×C4×D4×C43C2212yes—
43C22×C4⋊C4×C43C2211yescharacteristic
44C22×C22⋊C4×C43—21no—
45C22×C42×C43C2211yescharacteristic
46C22×D172C2311yescharacteristic
47C22×C22⋊D43—21no—
48C22×C4×D43—21no—
49C2×#10621882—24no—
50C2×#10621882—24no—
51C2×#10621882C2×C418yescomplemented
52C22×C43⋊C8C2312yes—
53C22×C43⋊C8C2312yes—
54#13063299—22no—
55#13063299C2×C414yescomplemented
56C4×D172—24no—
57C4×C22⋊D43—44no—
58C2×C4⋊Dic43C2311yescharacteristic
59C2×C4⋊Dic43C2311yescharacteristic
60C2×C4⋊Dic43C2311yescharacteristic
61C2×Dic43⋊C4—21no—
62C2×C22⋊Dic43—21no—
63C2×(C2×D43)⋊C4—21no—
64C2×(C2×D43)⋊C4—21no—
65C2×(C2×D43)⋊C4—21no—
66C2×(C2×D43)⋊C4—21no—
67C2×C4×Dic43—21no—
68#13063256—24no—
69#13063246—24no—
70#13063246—24no—
71#13063246C2×C418yescomplemented
72#13063245—24no—
73C22⋊(C43⋊C8)—22no—
74C22⋊(C43⋊C8)C2×C414yescomplemented
75#13063236—24no—
76C43⋊#95—21no—
77C43⋊#95—21no—
78C43⋊#95—21no—
79C25×C43—21no—
80C22×D4×C43—24no—
81C22×D4×C43—24no—
82C22×D4×C43—21no—
83C22×D4×C43—22no—
84C22×D4×C43C2×C414yescomplemented
85C22×D4×C43C2312yes—
86C23×C4×C43—28no—
87C23×C4×C43—21no—
88C23×C4×C43—22no—
89C23×C4×C43—22no—
90C23×C4×C43—44no—
91C23×C4×C43—44no—
92C23×C4×C43D411yescharacteristic
93C23×C4×C43D411yescharacteristic
94C23×C4×C43C2311yescharacteristic
95C4×D4×C43—28no—
96C4×D4×C43—28no—
97C4×D4×C43—24no—
98C4×D4×C43—24no—
99C4×D4×C43—24no—
100C4×D4×C43D418yescomplemented
101C2×C4⋊C4×C43—21no—
102C2×C4⋊C4×C43—24no—
103C2×C4⋊C4×C43C2312yes—
104C2×C22⋊C4×C43—28no—
105C2×C22⋊C4×C43—22no—
106C2×C22⋊C4×C43—22no—
107C2×C42×C43—24no—
108C2×C42×C43D411yescharacteristic
109C2×C42×C43D411yescharacteristic
110C2×C42×C43C2311yescharacteristic
111C2×C42×C43C2311yescharacteristic
112C24×C43—24no—
113C24×C43—24no—
114C24×C43—24no—
115C24×C43—22no—
116C24×C43—42no—
117C24×C43—42no—
118C24×C43—42no—
119C24×C43—42no—
120C24×C43C2×D411yescharacteristic
121C23×D43—21no—
122C2×D4×C43—24no—
123C2×D4×C43—216no—
124C2×D4×C43—24no—
125C2×D4×C43—216no—
126C2×D4×C43—24no—
127C2×D4×C43—22no—
128C2×D4×C43—22no—
129C2×D4×C43—22no—
130C2×D4×C43C22×C414yes—
131C2×D4×C43C2×D414yes—
132C2×D4×C43#4114yes—
133C22×C4×C43—22no—
134C22×C4×C43—22no—
135C22×C4×C43—22no—
136C22×C4×C43—22no—
137C22×C4×C43—24no—
138C22×C4×C43—28no—
139C22×C4×C43—22no—
140C22×C4×C43—28no—
141C22×C4×C43—22no—
142C22×C4×C43—22no—
143C22×C4×C43—22no—
144C22×C4×C43—41no—
145C22×C4×C43—42no—
146C22×C4×C43—432no—
147C22×C4×C43—44no—
148C22×C4×C43—44no—
149C22×C4×C43—44no—
150C22×C4×C43—44no—
151C22×C4×C43—42no—
152C22×C4×C43C22×C412yes—
153C22×C4×C43C2×D411yescharacteristic
154C22×C4×C43C2×D411yescharacteristic
155C22×C4×C43C2×D411yescharacteristic
156C22×C4×C43C2×D411yescharacteristic
157C22×C4×C43C2×D411yescharacteristic
158C22×C4×C43#4112yes—
159C22×C4×C43C2411yescharacteristic
160C2×C22⋊D43—24no—
161C2×C22⋊D43—42no—
162C2×C22⋊D43—42no—
163C22×Dic43—21no—
164C22×Dic43—21no—
165C22×Dic43—21no—
166C2×D172—21no—
167C2×D172—22no—
168C2×D172—22no—
169C2×D172C22×C412yescomplemented
170C2×C4×D43—42no—
171C2×C4×D43—42no—
172C2×C4×D43—42no—
173C4⋊C4×C43—24no—
174C4⋊C4×C43—22no—
175C4⋊C4×C43C2×D414yes—
176C22⋊C4×C43—28no—
177C22⋊C4×C43—28no—
178C42×C43—22no—
179C42×C43—22no—
180C42×C43—22no—
181C42×C43C2×D414yes—
182C22⋊Dic43—42no—
183#10621882—216no—
184(C2×D43)⋊C4—24no—
185(C2×D43)⋊C4—42no—
186(C2×D43)⋊C4—42no—
187(C2×D43)⋊C4—42no—
188C4⋊Dic43—21no—
189C4⋊Dic43—22no—
190C4⋊Dic43—22no—
191C4⋊Dic43C22×C412yescomplemented
192Dic43⋊C4—42no—
193C4×Dic43—42no—
194C2×C43⋊C8—22no—
195C2×C43⋊C8—22no—
196C2×C43⋊C8—22no—
197C2×C43⋊C8—22no—
198C2×C43⋊C8—22no—
199C2×C43⋊C8C22×C414yescomplemented
200C23×C43—21no—
201C23×C43—21no—
202C23×C43—21no—
203C23×C43—21no—
204C23×C43—216no—
205C23×C43—24no—
206C23×C43—24no—
207C23×C43—24no—
208C23×C43—48no—
209C23×C43—48no—
210C23×C43—48no—
211C23×C43—48no—
212C23×C43—48no—
213C23×C43—42no—
214C23×C43—42no—
215C23×C43—42no—
216C23×C43—42no—
217C23×C43—48no—
218C23×C43C4×D412yes—
219C23×C43AGL(1,ℤ/8)12yes—
220C23×C43AGL(1,ℤ/8)12yes—
221C23×C43C22×D411yescharacteristic
222C22×D43—22no—
223C22×D43—41no—
224C22×D43—41no—
225C22×D43—41no—
226C22×D43—41no—
227C22×D43—41no—
228C22×D43—41no—
229D4×C43—216no—
230D4×C43—24no—
231D4×C43—24no—
232D4×C43—24no—
233D4×C43C4×D418yescomplemented
234C2×C4×C43—28no—
235C2×C4×C43—21no—
236C2×C4×C43—21no—
237C2×C4×C43—21no—
238C2×C4×C43—21no—
239C2×C4×C43—21no—
240C2×C4×C43—21no—
241C2×C4×C43—21no—
242C2×C4×C43—21no—
243C2×C4×C43—24no—
244C2×C4×C43—24no—
245C2×C4×C43—21no—
246C2×C4×C43—24no—
247C2×C4×C43—21no—
248C2×C4×C43—21no—
249C2×C4×C43—44no—
250C2×C4×C43—416no—
251C2×C4×C43—416no—
252C2×C4×C43—48no—
253C2×C4×C43—42no—
254C2×C4×C43—42no—
255C2×C4×C43C4×D412yes—
256C2×C4×C43#12012yes—
257C2×C4×C43#12012yes—
258C2×C4×C43AGL(1,ℤ/8)12yes—
259C2×C4×C43C23×C411yescharacteristic
260C2×C4×C43C22×D412yes—
261C2×C4×C43C2×#4112yes—
262C2×C4×C43C2×#4111yescharacteristic
263C22⋊D43—48no—
264C2×Dic43—22no—
265C2×Dic43—41no—
266C2×Dic43—41no—
267C2×Dic43—41no—
268C2×Dic43—41no—
269C2×Dic43—41no—
270C2×Dic43—41no—
271C2×Dic43—41no—
272C2×Dic43—41no—
273D172—24no—
274D172—42no—
275C4×D43—84no—
276C43⋊C8—24no—
277C43⋊C8—42no—
278#21452—431nomaximal, Sylow
279C22×C43—22no—
280C22×C43—22no—
281C22×C43—22no—
282C22×C43—22no—
283C22×C43—22no—
284C22×C43—22no—
285C22×C43—28no—
286C22×C43—432no—
287C22×C43—44no—
288C22×C43—44no—
289C22×C43—44no—
290C22×C43—44no—
291C22×C43—44no—
292C22×C43—44no—
293C22×C43#44214yescomplemented
294C22×C43C2×C4×D411yescharacteristic
295C22×C43C2×#12011yescharacteristic
296C22×C43#52511yescharacteristic
297C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
298C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
299C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
300C22×C43#57511yescharacteristic
301C2×D43—42no—
302C2×D43—42no—
303C2×D43—42no—
304C2×D43—81no—
305C2×D43—81no—
306C2×D43—81no—
307C2×D43—81no—
308C4×C43—22no—
309C4×C43—22no—
310C4×C43—22no—
311C4×C43—22no—
312C4×C43—44no—
313C4×C43—41no—
314C4×C43C2×C4×D412yes—
315C4×C43#52512yes—
316Dic43—42no—
317Dic43—81no—
318Dic43—81no—
319C22×C4×D4D4311yescharacteristic, complemented
320C2×#447—431no—
321C2×#442—432no—
322C4×#121—431no—
323#1879—432no—
324#1874—431no—
325#1864—432no—
326#1853—431no—
327#1786—431no—
328#1762—432no—
329#1749—431no—
330C2×C43—24no—
331C2×C43—48no—
332C2×C43C4×#12011yescharacteristic
333C2×C43C2×#44211yescharacteristic
334C2×C43C4×AGL(1,ℤ/8)11yescharacteristic
335C2×C43#300311yescharacteristic
336C2×C43#300611yescharacteristic
337C2×C43#310911yescharacteristic
338C2×C43#311411yescharacteristic
339D43—82no—
340C23×D4C2×D4311yescharacteristic
341C24×C4—21no—
342C22×D8—432no—
343C2×#121—431no—
344C2×C4×D4—28no—
345C2×C4×D4—21no—
346C2×C4×D4—21no—
347C2×C4×D4—431no—
348C2×C4×D4—861no—
349C2×C4×D4C2×D4312yes—
350C2×C4×D4C2×D4312yes—
351C22×C4⋊C4C2×D4311yescharacteristic
352C22×C22⋊C4—21no—
353C22×C42C2×D4311yescharacteristic
354#447—438no—
355#442—868no—
356C2×#106—432no—
357C2×#102—432no—
358C2×#102—432no—
359C2×#102—432no—
360C2×#102—432no—
361C2×C22⋊C8—431no—
362C2×C22⋊C8—431no—
363C2×#97—432no—
364#390—431no—
365#389—431no—
366#388—861no—
367#386—861no—
368#380—431no—
369C4×C4⋊C4—431no—
370C4×C22⋊C4—861no—
371#340—432no—
372#340—432no—
373#336—432no—
374C43#2145211yescharacteristic, complemented, Sylow
375C25—21no—
376C22×D4—24no—
377C22×D4—24no—
378C22×D4—22no—
379C22×D4—21no—
380C22×D4—431no—
381C22×D4—861no—
382C22×D4C4×D4314yescomplemented
383C22×D4C22×D4312yes—
384C23×C4—28no—
385C23×C4—21no—
386C23×C4—22no—
387C23×C4—22no—
388C23×C4—44no—
389C23×C4—44no—
390C23×C4—861no—
391C23×C4C22⋊D4311yescharacteristic
392C23×C4C22⋊D4311yescharacteristic
393C23×C4C22×D4311yescharacteristic
394C2×D8—438no—
395C2×D8—864no—
396C2×D8—864no—
397C22×C8—432no—
398C22×C8—432no—
399#121—434no—
400#121—862no—
401C4×D4—28no—
402C4×D4—28no—
403C4×D4—24no—
404C4×D4—24no—
405C4×D4—24no—
406C4×D4—864no—
407C4×D4—1724no—
408C4×D4C22⋊D4318yescomplemented
409C2×C4⋊C4—21no—
410C2×C4⋊C4—24no—
411C2×C4⋊C4—431no—
412C2×C4⋊C4—431no—
413C2×C4⋊C4—431no—
414C2×C4⋊C4—861no—
415C2×C4⋊C4C22×D4312yes—
416C2×C22⋊C4—28no—
417C2×C22⋊C4—22no—
418C2×C22⋊C4—22no—
419C2×C22⋊C4—861no—
420C2×C22⋊C4—861no—
421C2×C22⋊C4—861no—
422C2×C22⋊C4—861no—
423C2×C22⋊C4—861no—
424C2×C42—24no—
425C2×C42—861no—
426C2×C42C22⋊D4311yescharacteristic
427C2×C42C22⋊D4311yescharacteristic
428C2×C42C22×D4311yescharacteristic
429C2×C42C22×D4311yescharacteristic
430#106—864no—
431#102—438no—
432#102—864no—
433#102—864no—
434#102—864no—
435C22⋊C8—434no—
436C22⋊C8—862no—
437#97—864no—
438#95—861no—
439#95—861no—
440#95—861no—
441C24—24no—
442C24—24no—
443C24—24no—
444C24—22no—
445C24—42no—
446C24—42no—
447C24—42no—
448C24—42no—
449C24—861no—
450C24C2×C22⋊D4311yescharacteristic
451C2×D4—24no—
452C2×D4—24no—
453C2×D4—216no—
454C2×D4—216no—
455C2×D4—24no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—432no—
460C2×D4—864no—
461C2×D4—862no—
462C2×D4—862no—
463C2×D4—861no—
464C2×D4—1722no—
465C2×D4—1722no—
466C2×D4C2×C4×D4314yes—
467C2×D4#1062189814yes—
468C2×D4C2×C22⋊D4314yes—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—24no—
475C22×C4—28no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—28no—
480C22×C4—41no—
481C22×C4—432no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—44no—
488C22×C4—861no—
489C22×C4—861no—
490C22×C4—861no—
491C22×C4—1722no—
492C22×C4—1722no—
493C22×C4—1722no—
494C22×C4C2×C4×D4312yes—
495C22×C4#1062189812yes—
496C22×C4C2×C22⋊D4311yescharacteristic
497C22×C4C2×C22⋊D4311yescharacteristic
498C22×C4C2×C22⋊D4311yescharacteristic
499C22×C4C2×C22⋊D4311yescharacteristic
500C22×C4C2×C22⋊D4311yescharacteristic
501C22×C4C23×D4311yescharacteristic
502D8—8616no—
503C2×C8—434no—
504C2×C8—862no—
505C2×C8—862no—
506C2×C8—862no—
507C2×C8—862no—
508C2×C8—862no—
509C4⋊C4—24no—
510C4⋊C4—22no—
511C4⋊C4—432no—
512C4⋊C4—861no—
513C4⋊C4—862no—
514C4⋊C4—862no—
515C4⋊C4—1722no—
516C4⋊C4C2×C22⋊D4314yes—
517C22⋊C4—28no—
518C22⋊C4—28no—
519C22⋊C4—864no—
520C22⋊C4—1722no—
521C22⋊C4—1722no—
522C22⋊C4—1722no—
523C22⋊C4—1722no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—1722no—
528C42C2×C22⋊D4314yes—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—216no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—48no—
538C23—48no—
539C23—48no—
540C23—48no—
541C23—48no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—48no—
547C23—862no—
548C23—1721no—
549C23—1721no—
550C23—1721no—
551C23—1721no—
552C23—1721no—
553C23—1721no—
554C23C4×C22⋊D4312yes—
555C23#1306335012yes—
556C23#1306335112yes—
557C23C22×C22⋊D4311yescharacteristic
558D4—216no—
559D4—24no—
560D4—24no—
561D4—24no—
562D4—864no—
563D4—1728no—
564D4—1722no—
565D4C4×C22⋊D4318yescomplemented
566C2×C4—28no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—21no—
578C2×C4—24no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—44no—
582C2×C4—416no—
583C2×C4—416no—
584C2×C4—48no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—862no—
588C2×C4—3444no—
589C2×C4—1721no—
590C2×C4—1721no—
591C2×C4—1721no—
592C2×C4—1721no—
593C2×C4—1721no—
594C2×C4—1721no—
595C2×C4—1721no—
596C2×C4—1721no—
597C2×C4C4×C22⋊D4312yes—
598C2×C4#1306329412yes—
599C2×C4#1306329412yes—
600C2×C4#1306335112yes—
601C2×C4C22×C4×D4311yescharacteristic
602C2×C4C22×C22⋊D4312yes—
603C2×C4C2×#1062189812yes—
604C2×C4C2×#1062189811yescharacteristic
605C8—864no—
606C8—1722no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—28no—
614C22—432no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—3441no—
622C22—3441no—
623C22—3441no—
624C22—3441no—
625C22—1722no—
626C22—1722no—
627C22—1722no—
628C22#1458300914yescomplemented
629C22C2×C4×C22⋊D4311yescharacteristic
630C22C2×#1306329411yescharacteristic
631C22#1458341011yescharacteristic
632C22C2×#1306335011yescharacteristic
633C22C2×#1306335111yescharacteristic
634C22C2×#1306335111yescharacteristic
635C22#1458369711yescharacteristic
636C4—22no—
637C4—22no—
638C4—22no—
639C4—22no—
640C4—44no—
641C4—41no—
642C4—3441no—
643C4—3441no—
644C4—1722no—
645C4C2×C4×C22⋊D4312yes—
646C4#1458341012yes—
647C2—24no—
648C2—48no—
649C2—3442no—
650C2C4×#1306329411yescharacteristic
651C2C2×#1458300911yescharacteristic
652C2C4×#1306335111yescharacteristic
653C2#1914382211yescharacteristic
654C2#1914384711yescharacteristic
655C2#1914468611yescharacteristic
656C2#1914473311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D43
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC22×C4×D4×C43C2
socleC23×C43C22×D4
prime radicalC2×#14583033C2
cosocleC22×C4×C43C24
hypercenterC22×C4×D4D43
hypocenterC43#21452
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×D4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#14583009C2semidirect
3. C4×#13063299C2semidirect
4. #19134943C2semidirect
5. #19134912C2semidirect
6. #19134856C2semidirect
7. #19134788C2semidirect
8. #19134526C2semidirect
9. #19134420C2semidirect
10. #19134357C2semidirect
11. C22×C4×D4×C43C2semidirect
14. C2×C4×D172C22semidirect
16. #14583033C4semidirect
18. C2×#13063256C22semidirect
22. C2×#13063245C22semidirect
25. C2×#13063236C22semidirect
26. #14582801C22semidirect
27. #14582797C22semidirect
31. C4×C4⋊Dic43C22semidirect
33. #14582661C22semidirect
34. #14582661C22semidirect
35. #14582654C22semidirect
51. C2×#10621882C2×C4semidirect
55. #13063299C2×C4semidirect
71. #13063246C2×C4semidirect
74. C22⋊(C43⋊C8)C2×C4semidirect
84. C22×D4×C43C2×C4semidirect
100. C4×D4×C43D4semidirect
169. C2×D172C22×C4semidirect
191. C4⋊Dic43C22×C4semidirect
199. C2×C43⋊C8C22×C4semidirect
233. D4×C43C4×D4semidirect
293. C22×C43#442semidirect
319. C22×C4×D4D43semidirect
374. C43#21452semidirect
382. C22×D4C4×D43semidirect
408. C4×D4C22⋊D43semidirect
565. D4C4×C22⋊D43semidirect
628. C22#14583009semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#145830332
C2×#145830092
C4×#130632992
#191349432
#191349122
#191348562
#191347882
#191345262
#191344202
#191343572
C22×C4×D4×C432
#2145243